A living system with autonomously emerging boundaries was presented in previous work. This paper gives a precise mathematical construction of this system by a classification of the flow diagrams and rigorous definition of the global inverse rules of cellular automata. Our model is based on the macroscopic and microscopic perpetual disequilibrations which were first introduced and studied by Mastuno. Because of the perpetual disequilibrations, the boundary conditions are determined a posteriori in succession. This means that our biological model contains the uncontrollability of both the macroscopic and the microscopic boundary conditions. Furthermore, in a typical case of this model, the local transition rule is perpetually changed because of the uncontrollable boundary conditions. Thus, an interaction between the dynamics and boundary conditions which plays an important role in living systems appears in this autonomous artificial life.
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